L(s) = 1 | + 2.54i·2-s − 2.15i·3-s − 4.48·4-s + (−0.817 + 2.08i)5-s + 5.48·6-s + 2.93i·7-s − 6.31i·8-s − 1.63·9-s + (−5.29 − 2.08i)10-s − 0.635·11-s + 9.64i·12-s − 7.48·14-s + (4.48 + 1.76i)15-s + 7.11·16-s + 1.22i·17-s − 4.16i·18-s + ⋯ |
L(s) = 1 | + 1.80i·2-s − 1.24i·3-s − 2.24·4-s + (−0.365 + 0.930i)5-s + 2.23·6-s + 1.11i·7-s − 2.23i·8-s − 0.545·9-s + (−1.67 − 0.658i)10-s − 0.191·11-s + 2.78i·12-s − 1.99·14-s + (1.15 + 0.454i)15-s + 1.77·16-s + 0.296i·17-s − 0.981i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.258973 - 0.379964i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.258973 - 0.379964i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.817 - 2.08i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 2.54iT - 2T^{2} \) |
| 3 | \( 1 + 2.15iT - 3T^{2} \) |
| 7 | \( 1 - 2.93iT - 7T^{2} \) |
| 11 | \( 1 + 0.635T + 11T^{2} \) |
| 17 | \( 1 - 1.22iT - 17T^{2} \) |
| 19 | \( 1 + 1.36T + 19T^{2} \) |
| 23 | \( 1 - 2.15iT - 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 + 1.22iT - 37T^{2} \) |
| 41 | \( 1 + 9.96T + 41T^{2} \) |
| 43 | \( 1 + 1.36iT - 43T^{2} \) |
| 47 | \( 1 - 6.16iT - 47T^{2} \) |
| 53 | \( 1 - 0.642iT - 53T^{2} \) |
| 59 | \( 1 + 7.59T + 59T^{2} \) |
| 61 | \( 1 + 2.27T + 61T^{2} \) |
| 67 | \( 1 + 8.03iT - 67T^{2} \) |
| 71 | \( 1 - 2.63T + 71T^{2} \) |
| 73 | \( 1 + 10.3iT - 73T^{2} \) |
| 79 | \( 1 + 1.03T + 79T^{2} \) |
| 83 | \( 1 - 11.8iT - 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 97 | \( 1 - 14.7iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.74173392952239168676849490448, −9.457797745381669568048547583397, −8.669376790941399349731989249005, −7.77967925539277369347480234338, −7.40810622291495769416880545234, −6.48422215315141754915077177356, −6.02735376188529160871096488171, −5.07388071330617749330035037157, −3.61350654228351994489530528149, −2.10258227043589060187693926370,
0.22533815523552216788235204707, 1.66032282059052556281310904638, 3.31977278812437142183667566567, 3.98127407578818119330710396704, 4.60512530958068638995598183598, 5.34488862187639909150622051578, 7.29298557928498136597821374324, 8.507158343299605554870624715137, 9.145407580376321236085649748330, 9.916177041083688148832793636799